Shape instability on swelling of a stretched nematic elastomer filament

Liquid crystalline elastomers combine the ordering properties of liquid crystals with elasticity of crosslinked polymer networks. In monodomain (permanently aligned) elastomers, altering the orientational (nematic) order causes changes in the equilibrium sample length, which is the basis of the famous effect of large-amplitude reversible mechanical actuation. The stimulus for this effect could be a change in temperature, or illumination by light in photosensitized elastomers, but equally the nematic order changes by mixing with a solvent. This work theoretically investigates a competition between the spontaneous contraction on swelling of a monodomain nematic elastomer and the externally imposed stretching. We find that this competition leads to bistability in the system and allows a two-phase separation between a nematic state with lower swelling and an isotropic state with higher solvent concentration. We calculated the conditions in which the instability occurs as well as the mechanical and geometric parameters of equilibrium states. Being able to predict how this instability arises will provide opportunities for exploiting nematic elastomer filaments.

Figure 1

(a) The nematic order parameter $Q$ as a function of scaled temperature $T/T_c$, the data from two recent experimental studies highlighting the critical and supercritical cases. (b) The order parameter plotted against solvent concentration, labeled here as ${\varphi }_{\mathrm{imp}}$, using the parametrization in Eqs. (5) and (6).

Figure 2

(a) The spontaneous extension of a uniaxial nematic elastomer along its director as a function of temperature, Eqs. (4) and (5). (b) The spontaneous extension in the case when the nematic order is reduced by the added solvent of volume fraction $\varphi$, Eq. (7). The dashed line is the isotropic $\lambda ={(1-\varphi )}^{-1/3}$.

Figure 3

Plots of $F_{\text{el}}, F_{\text{mix}}$, and $F_{\text{tot}}$ against solvent volume fraction $\varphi$ in good solvent, for $\lambda$ = 1, that is, without initial prestretching of the dry elastomer. The kink seen in the elastic part of the free energy is a point where the nematic order parameter drops to zero; cf. Fig. 1.

Figure 4

Schematic illustration of the instability: (a) aligned elastomer in stress-free equilibrium nematic state; (b) the sample stretched by extension $\lambda$ along the director axis, after which the length $L$ remains fixed; (c) the sample is swollen by adding solvent molecules with volume fraction $\varphi$. (d) After the instability the sample is split into two regions with different $\varphi$ and $\lambda$, in proportion $x$ and ($1-x$); the higher-$\varphi$ portion is in the isotropic state.

Figure 5

(a) Plot of $F_{\text{tot}}$ against $\varphi$ for different values of $\chi$, at $\lambda$ = 1 (i.e., without initial prestretching of the dry elastomer). Lower values of $\chi$ mean better solvent quality and stronger natural swelling. The step, also seen in Fig. 3, is a point where the nematic phase turns isotropic with an apparent singularity; cf. Fig. 1. (b) Plot of $F_{\text{tot}}$ against $\varphi$ for different values of $\lambda$ with $\chi$ = 0. ${\varphi }_1$ and ${\varphi }_2$ are the values of $\varphi$ at the nematic and isotropic minima, respectively. Dash line denotes when the free energy values are equal at the two minima (which we denote ${\lambda }_{\text{crit}}$).

Figure 6

Plot of ${\chi }_c$, below which bistability occurs, against the imposed extension $\lambda$. Note the very weak dependence ${\chi }_c\left(\lambda \right)$ over a long range of strains.

Figure 7

Plot of ${\lambda }_{\text{crit}}$, at which the total free energy of the nematic and isotropic states are equal, against mixing parameter $\chi$. The dashed line marks the nonstretched state ($\lambda =1$), corresponding to Fig. 5.

Figure 8

The energy cost for the instability, $\mathrm{\Delta }F$, as a function of separation fraction $x$ for different stretching $\lambda$. The three plots are for an increasing crosslinking density of elastomer network (see below): (a) $\nu$ = 0.5, i.e., about 20 monomers between crosslinks, (b) $\nu$ = 0.1, about 10 monomers, and (c) $\nu$ = 0.2, i.e., every fifth monomer is crosslinked. The minima of curves denote the optimum molar fraction of the separation $x_c$.

Figure 9

Plot of $x_c$ as a function of $\lambda$ for different crosslinking ratios $\nu$. When $x_c\ne 1$ or 0, the fiber is split into two distinct coexisting regions as shown in Fig. 4. The pure states: $x_c$ = 0 implies the homogeneous isotropic state; $x_c$ = 1 implies the homogeneous nematic state. It appears the 10% crosslinking density is the “best” case to observe the coexistence and swelling instability.

Figure 10

The overall swelling ratio of the sample as a function of crosslinking ratio $\nu$ for the imposed strain $\lambda$ = 1 (no stretching) and $\lambda$ = 2. Dashed lines show the swelling ratio if the elastomer remained in the homogeneous isotropic or nematic state, respectively.